Well I can tell you it's NOT a dead battery.
I have the exact same problem with my old Macbook Core Duo.
I have several old batteries and a couple of new ones, and all cause the same behavior.
1) All batteries will charge
2) Adapter LED goes from orange to green, and the batteries peak at 97-100%
3) Macbook will only turn on when plugged in.
4) If not plugged in to AC, Macbook will spin up the drive, no chime, then spin down.
Of what I can find online, it's either something wrong with the battery connector circuitry, although everything, even coconut battery utility, says everything is within normal limits. Or it's the logic board.